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AP STATISTICS
LESSON 10.1
INTRODUCTION TO INFERENCE
ESSENTIAL QUESTION:
What is a confidence interval and
how are a they used to make
inferences?
Objectives:
 To find confidence intervals.
 To interpret the meaning of confidence
intervals.
Introduction
Often we are not content with information
about the sample.
We want to infer from the sample data
some conclusion about a wider population
that the sample represents.
Statistical Inference
Statistical inference provides methods for
drawing conclusions about a population from
sample data.
We use probability to express the strength of
our conclusions. Probability allows us to take
chance variation into account and so to
correct our judgment by calculations.
Example 10.1
page 536
Draft Lotteries and Drug Studies
In the Vietnam War years, a lottery determined
the order in which men were drafted for army
service. The lottery assigned draft numbers by
choosing birth dates in random order.
We expect a correlation of about zero. The
actual correlation between birth date and draft
number in the first draft lottery was r = - .226
That is men born later in the year tended to get
lower draft numbers. The probability of the
correlation being that far from 0 by chance is
0.001 in a truly random lottery.
Estimating With Confidence
A computer will do the arithmetic, but you must still exercise
judgment based on understanding.
The methods of formal inference require the long-run
regular behavior that probability describes.
Inference is most reliable when the data are produced by a
properly randomized design.
When you use statistical inference you are acting as if the
data are random sample or come from a randomized
experiment.
If this is not true, your conclusions may be open to
challenge.
Example 10.1 (continued…)
Suppose that we know that the standard
deviation of SAT math is σ = 100
σ/√n = 100/√500 = 4.5 x = 463
Inference about the unknown μ starts from this
sample distribution. Figure 10.1 shows different
SRS of 500 California seniors and a graph of
their distribution.
Example 10.2 page 536
SAT Math Scores in California
In 2000, 1,260,278 college bound seniors took the SAT. Their mean
SAT math score was 514 with a standard deviation of 113. For the SAT
verbal, the mean was 505 with a standard deviation of 111.
Suppose you want to estimate the mean SAT math score
for the more than 350,000 high school seniors in
California. Only 49% of the California students take the
SAT. These self-selected seniors are planning to attend
college and so are not representative of all college
seniors. You give the test to an SRS of 500 high school
seniors and determine mean x = 461.
What can you say about the mean score μ in the
population of all 350,000 seniors?
Essential Facts About Sampling
Distribution of x.

The central limit theorem tells us that the
mean x of 500 scores has a distribution that is
close to normal.
 The mean of this normal sampling distribution
is the same as the unknown mean μ of the
entire population.
 The standard deviation of x for an SRS of 500
students is σ/ √ 500, where σ is the standard
deviation of individual SAT math scores
among all California high school seniors.
Statistical Confidence
The 68-95-99.7 rule says that in 95% of all
samples, the mean score x for the sample will be
within two standard deviations of the population
mean score μ. So the mean x of 500 SAT math
scores will be within 9 points of μ in 95% of all
samples.
 Whenever x is within 9 points of the unknown μ, μ
is within 9 points of the observed x. This happens
in 95% of all samples.
 So in 95% of all samples, the unknown μ lies
between x – 9 and x + 9.

Figures 10.2 and 10.3
Page 539
Example 10.3
page 540
95% Confidence
Our sample of 500 California seniors gave X = 461.
We say that we are 95% confident that the unknown SAT
math sore for all California high school seniors lies between
452 and 470.
Be sure you understand the grounds for our confidence.
There are only two possibilities:
1.
2.
The interval between 452 and 470 contains the true μ.
Our SRS was one of the few samples for which x is not within 9
points of the true μ. Only 5% of all samples give such inaccurate
results.
Example 10.3 (continued…)
Margin of Error - The margin of error ±
9 shows how accurate we believe our
guess is, based on the variability of the
estimate.
This is a 95% confidence interval
because it catches the unknown μ in
95% of all possible samples.
Confidence interval
A level C confidence interval for a parameter has
two parts:
1. An interval calculated from the data, usually of
the form
estimate ± margin of error
2. A confidence level C, which gives the
probability that the interval will capture the true
parameter value in repeated samples.
(Use decimal form for %.)
Figure 10.4
Page 541
The graph
shows that
only one
interval does
not contain
parameters
mean.